 I'm Zor. Welcome to Unisor Education. Today's topic is linear equations. Well, linear equations are really simple. There is nothing much which we can really discuss in this particular lecture. Still, I will try to present something which might be interesting. I doubt it will be new, but at least interesting about linear equations. So, what is a linear equation? Obviously, this is something of this type. The first question which comes up when you are talking about any equation actually, including linear, is what's the domain where we would like to look for solutions of this particular equation? As you understand, it can be different, and certain equations might or might not have solutions in this or that domain. For instance, if I'm looking for a solution to this equation in the domain of integer numbers, which means I'm looking for an integer x, which is multiplied by 2 plus 3 gives 0, there are enough solutions here. If, however, I'm looking for a solution of this equation in the domain of rational numbers, then yes, obviously x equals to minus 3 second is a solution, but this is the rational number. So, in the domain of rational numbers, there is a solution. Now, what if I will ask about this particular equation, and I will tell that I'm looking for a solution in the domain of rational numbers? Well, we all know that square root of 2 is irrational number, so there is no solution in the area in the domain of rational numbers in this particular case. But in the domain of real numbers, the solution is and x is equal to minus square root of 2. And obviously, the same can be actually said about complex numbers. If I'm looking for a solution to 3 plus i times x minus 2 times i minus 7 equals to 0, well, obviously the solution to this particular equation does not exist in the domain of real numbers, but it does exist in the domain of complex numbers. Okay, so this was just an illustration of certain I would say peculiar things which might happen with solutions, and actually there are quite a number of very interesting problems related to this particular topic. For instance, you can have the problem solve this particular equation in the domain of let's say integer numbers. So that actually presents a certain problem. I'm not talking obviously about linear solutions, they are easy, but there are some more complicated equations where this definitely can be applied. Alright, so one important topic right now is that we always have to consider what's the domain of our equation. Now for this particular lecture, I will always use the domain of real numbers. For different reasons primarily because it's most frequently occurred case and also I would like to present certain graphical representation and graphical representation is obviously related to real numbers because real numbers are mapped exactly to all the points on the line on x-axis and the y-axis. Alright now, is any equation of this type is a linear equation? Well strictly speaking no, because we have to put a condition a is not equal to zero. Now, we will consider separately a case when a is equal to zero a little bit later but right now we should assume that a is not equal to zero also we assume that both a and b are real numbers and x belongs to domain of real numbers. So these are assumptions which I'm staging right now on the onset and keep it in mind and now we will go with easy solution to this equation in this particular case. Obviously we have to apply transformations as usually to solve the equation and the first transformation should be I am converting both sides by subtracting b. Transformation which corresponds to each number, that number minus b now if I will apply this transformation to the left part I will obviously get this and right part will be this one. This is obviously reduced so we have a times x equals to minus b. Next transformation again quite obvious, I divide by a. Now that's exactly where a not equal to zero comes handy because otherwise I wouldn't be able to do this. So applying this transformation I'm getting x equals to minus b over a. That's it, no more than that. Simple straightforward without any kind of losing solutions, gaining solutions because all the transformations are invariant considering a is not equal to zero as we have assumed from the very beginning. So everything is okay in this particular case. Now our next job is to represent this particular equation and its solution geometrically using code and the taxes. So let me just write down only the solution to this equation x is equal to minus b over a and let's go to graphical representation. Now let's start with y is equal to x and next we will go to y is equal to a times x and next we will go to a times x plus b. So these are three steps which we are going to make. Well y is equal to x, everybody knows what that is. This is line which bisects this straight angle and this is 45 degrees so we all know that this is y equals to x y because obviously for every x if I would go up the length of this will be exactly the same as the length of this. If this length is x and this length is y then y is equal to x. So for every point here this is an obvious property. Okay now what happens if I multiply it by a? Well let's consider two cases, three cases actually. a is equal to zero which we have assumed in the very beginning which is not but here I would like actually to present the graph with a equals to zero and then the positive a and the negative a. With positive a it's actually quite simple because right now if this length and this length are exactly the same so this is x but now I don't want x, I want a times x which means this length becomes a times longer if a is greater than one or shorter is if a is less than one. So we are right now assumed that a is positive. Let's say a is equal to two. Well the graph will be twice as steep so instead of this length I will get up to this and the function will be like this so this is y is equal to two x. Now if my a is let's say one half this segment becomes smaller it will be half of this and the function will be graph will be like this so this is y is equal to one half x and now you understand basically the principle the larger a the steeper this particular line goes and the smaller a it goes more horizontally and here is when a is equal to zero when a is equal to zero it just coincides with this line so this is y equals zero times x which means for any x y is equal to zero so every point on this particular x axis is actually a representation of the graph y is equal to zero times x or y is equal to zero basically. Now let's move further down if a is negative well quite obviously if a is equal to let's say minus one so y is equal to minus x it will bisect this angle so this will be y is equal to minus x and same thing here this will be y is equal to minus one half x and a steeper one will be y is equal to two x alright so we have all the different angles depending on the a and I would like actually to have only one we have too many lines here you understand the point so I will just draw one particular line with one particular value of a I'll choose the positive a because it's easier so again this is my graph and this is y is equal to a times x this is zero this is x axis this is y axis alright now so we finished with this one how about this well obviously again every y is getting lifted by b well up if b is positive and down if b is negative let's assume b is positive and then the whole thing becomes like this where this particular point on the y axis is basically the b we took this line and shifted it up by b so this particular segment is always b so this represents the function a is equal to y is equal to ax plus b that's exactly what we wanted to build and now when we have actually when we have this graph now we can solve graphically the equation ax plus b is equal to zero if this is a y is equal to ax plus b question is when y is equal to zero because when y is equal to zero that actually means that ax plus b is equal to zero so under what conditions y is equal to zero well obviously if this is all the different values of y then y is equal to zero is this one so this particular point represents y is equal to zero and question is what is x in this particular case well again let's make a slide with better graph not better I'll just clean up this piece which we don't need anymore ok now we know that this particular segment is equal to b now this particular segment is that particular x which brings y to zero so what's the ratio between this segment and this segment well let's put point m this is b and this is point m so what I'm interested right now is what is all m ratio to all m well remember when a was equal to one it was bisecting the line was bisecting the straight angle and these two segments were always equal to each other and for any a the line actually stretched upwards so basically what it represents it represents that the ratio becomes instead of being one in case a is equal to one it becomes whatever a value is so this is always a that what it means to stretch it vertically by a if you multiply a by x so in this particular case since the ratio between this segment and this segment is a now all m is equal to b and all n is equal to that particular x which is a solution to our equation when ax plus b is equal to zero well obviously from here and x is equal to b divided by a now we have here a slight difference between this and this now why is this such a case because you see x is actually to the left of the zero so I shouldn't really say it's x it's the absolute value of x here and b again is considered positive so I have to really use only positive values here if we are talking about geometry but in real life if you will take a look at this particular b and this particular x you see that x is negative and b is positive so that's why we will get rid of all these absolute values my x is really negative and to get it positive I have to multiply it by minus one and that actually corresponds to this formula so this is a graphical solution in case a is positive b is positive x actually becomes negative in this case so a is positive because we have it steeped that way because if a is negative it will be a way around and b is positive it means we shifted it upwards and that would shift the solution to the left of the zero to the negative values so that's basically the graphical representation of the solution now let's go back to a small case when a is equal to zero I promise to consider it and I will do it graphically right now again this is not a real linear equation however from the general approach to use the graphics for solving these particular equations you can save the following what is y is equal to zero times x plus d well you remember that y is equal to zero times x which is zero is the horizontal line which coincides with the x axis and if you add b to this it will be the line which is parallel and the b is this particular segment ok now what is the value of x which brings y to zero well as you see we never cross the x axis which means y is never equal to zero which means that if b is not equal to zero if b is positive or negative it basically means that there are no solutions here so obviously if b is not equal to zero there is no such y which can bring it to zero and finally if b does if b is equal to zero we have a solution actually of this particular equation now think about this this is the line which is exactly coincides with x axis which means it's always equal to zero so no matter what x is y is always equal to zero which means we have an infinite number of solutions any x is basically a solution so if a is equal to zero we have either no solutions at all if b is not equal to zero or infinite number of solutions when b is equal to zero so it's not really an interesting point to consider about this particular case so that's why in the very beginning at the onset we are stating that a is not equal to zero to make it a real equation that's what actually is important real equation always has a solution in this particular case when a is not equal to zero and it's only one solution well that concludes this exercise about linear equations thank you very much